Abstract
A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L1 ∩ BV class was constructed in [1]. In the current paper, we will continue to study the uniqueness and regularity of the constructed solution. The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables. With these finer estimates, we obtain higher order regularity of the constructed solution to Navier-Stokes equation, so that all of the derivatives in the equation of conservative form are in the strong sense. Moreover, this regularity also allows us to identify a function space such that the stability of the solutions can be established there, which eventually implies the uniqueness.
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References
Liu T P, Yu S H. Navier-Stokes equations in gas dynamics: Green’s function, singularity, and well-posedness. Comm Pure Appl Math, 2022, 75(2): 223–348
Smoller J. Shock Waves and Reaction-Diffusion Equations. Berlin: Springer, 1994
Nash J. Le problème de Cauchy pour les équations différentielles d’un fluide général. Bulletin de la Soc Math de France (in French), 1962, 90: 487–497
Nash J. Continuity of solutions of parabolic and elliptic equations. Amer J Math, 1958, 80(4): 931–954
Itaya N. On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid. Kodai Math Sem Rep, 1971, 23: 60–120
Kanel’Ya I. On a model system of equations for one-dimensional gas motion. Diff Uravn (in Russian), 1968, 4: 374–380
Kazhikhov A V, Shelukhin V V. Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl Mat Meh (in Russian), 1977, 41(2): 282–291
Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat conductive gases. J Math Kyoto Univ, 1980, 20(1): 67–104
Shizuta Y, Kawashima S. Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math J, 1985, 14: 249–275
Kawashima S. Large-time behavior of solutions to hyperbolic parabolic systems of conservation laws and applications. Proceedings of the Royal Society of Edinburgh, 1987, 106: 169–194
Hoff D. Global existence for 1D compressible isentropic Navier-Stokes equations with large initial data. Trans Amer Math Soc, 1987, 303: 169–181
Huang X D, Li J. Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations. Arch Ration Mech Anal, 2018, 227(3): 995–1059
Jiu Q S, Li M J, Ye Y L. Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data. J Differential Equations 2014, 257: 311–350
Jiu Q S, Wang Y, Xin Z P. Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum. J Math Fluid Mech, 2014, 16: 483–521
Mellet A, Vasseur A. Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J Math Anal, 2008, 39: 1344–1365
Hoff D. Discontinuous solutions of the Navier-Stokes equations for compressible flow. Arch Rational Mech Anal, 1991, 114: 15–46
Hoff D. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J Differential Equations, 1995, 120: 215–254
Lions P L. Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models. New York: Oxford University Press, 1996
Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford: Oxford University Press, 2004
Chen G Q, Hoff D, Trivisa K. Global solutions of the compressible navier-stokes equations with larger discontinuous initial data. Commun Partial Diff Equations, 2000, 25(11): 2232–2257
Dafermos C M, Hsiao L. Development of singularities in solutions of the equations of nonlinear thermoe-lasticity. Quart Appl Math, 1986, 44: 462–474
Hsiao L, Jiang S. Nonlinear hyperbolic-parabolic coupled systems. Handbook of Differential Equations: Evolutionary Equations, 2002, 1: 287–384
Novotnỳ A, Strašsraba I. Introduction to the Mathematical Theory of Compressible Flow. Oxford: Oxford University Press, 2004
Liu T P. Pointwise convergence to shock waves for viscous conservation laws. Comm Pure Appl Math, 1997, 50(12): 1113–1182
Liu T P, Yu S H. The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Comm Pure Appl Math, 2004, 57(12): 1542–1608
Liu T P, Yu S H. Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Comm Pure Appl Math, 2007, 60(3): 295–356
Liu T P, Zeng Y. Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws. Providence, RI: Amer Math Soc, 1997
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The authors declare no conflict of interest.
This work was partially supported by the National Key R&D Program of China (2022YFA1007300); Wang was supported by the NSFC (11901386, 12031013) and the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA25010403); Zhang was supported by the NSFC (11801194, 11971188) and the Hubei Key Laboratory of Engineering Modeling and Scientific Computing.
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Wang, H., Zhang, X. The Regularity and Uniqueness of a Global Solution to the Isentropic Navier-Stokes Equation with Rough Initial Data. Acta Math Sci 43, 1675–1716 (2023). https://doi.org/10.1007/s10473-023-0415-x
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DOI: https://doi.org/10.1007/s10473-023-0415-x